| L(s) = 1 | + 10·5-s + 98·7-s − 54·11-s + 240·13-s − 1.90e3·17-s − 400·19-s − 2.93e3·23-s − 1.89e3·25-s − 5.54e3·29-s + 4.90e3·31-s + 980·35-s + 2.95e3·37-s − 6.07e3·41-s + 3.30e3·43-s − 1.69e4·47-s + 7.20e3·49-s − 6.89e3·53-s − 540·55-s − 5.38e4·59-s + 4.14e3·61-s + 2.40e3·65-s + 3.35e4·67-s − 7.35e4·71-s + 128·73-s − 5.29e3·77-s + 5.77e4·79-s − 2.30e4·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s + 0.755·7-s − 0.134·11-s + 0.393·13-s − 1.59·17-s − 0.254·19-s − 1.15·23-s − 0.606·25-s − 1.22·29-s + 0.916·31-s + 0.135·35-s + 0.354·37-s − 0.563·41-s + 0.272·43-s − 1.12·47-s + 3/7·49-s − 0.337·53-s − 0.0240·55-s − 2.01·59-s + 0.142·61-s + 0.0704·65-s + 0.912·67-s − 1.73·71-s + 0.00281·73-s − 0.101·77-s + 1.04·79-s − 0.366·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 p T + 1994 T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 54 T + 284302 T^{2} + 54 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 240 T + 739862 T^{2} - 240 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1906 T + 1860002 T^{2} + 1906 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 400 T + 3279798 T^{2} + 400 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2930 T + 9774686 T^{2} + 2930 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5540 T + 40407182 T^{2} + 5540 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4904 T + 59914302 T^{2} - 4904 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2952 T + 138794486 T^{2} - 2952 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6070 T + 232254602 T^{2} + 6070 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3304 T + 197837766 T^{2} - 3304 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16988 T + 429309854 T^{2} + 16988 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6896 T + 613869254 T^{2} + 6896 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 53820 T + 1699029142 T^{2} + 53820 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 68 p T + 1287381294 T^{2} - 68 p^{6} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 33516 T + 2261444678 T^{2} - 33516 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 73518 T + 4934300734 T^{2} + 73518 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 128 T - 360358674 T^{2} - 128 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 57740 T + 6120687198 T^{2} - 57740 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 23016 T - 4303421546 T^{2} + 23016 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 141530 T + 14809201898 T^{2} - 141530 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 226216 T + 29691907182 T^{2} + 226216 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.698621357691645855154396779744, −9.632426649691190045388261930508, −8.924595907220051428186947398244, −8.605047807448879723279010070588, −7.999466963441445661836865760849, −7.85000536979697586208624090541, −7.19640931536114569642355592199, −6.61340208993397008404142752301, −6.13989783238775839149158436717, −5.88644039374264784313795615549, −4.95751104261027156037606843425, −4.84544816534123082318780079736, −3.97798755369182886633256897094, −3.85924955071786598137556514679, −2.83323491538201949059364176099, −2.30328644003832381056158689407, −1.72523941582283573050028917718, −1.26019755765368447561610254117, 0, 0,
1.26019755765368447561610254117, 1.72523941582283573050028917718, 2.30328644003832381056158689407, 2.83323491538201949059364176099, 3.85924955071786598137556514679, 3.97798755369182886633256897094, 4.84544816534123082318780079736, 4.95751104261027156037606843425, 5.88644039374264784313795615549, 6.13989783238775839149158436717, 6.61340208993397008404142752301, 7.19640931536114569642355592199, 7.85000536979697586208624090541, 7.999466963441445661836865760849, 8.605047807448879723279010070588, 8.924595907220051428186947398244, 9.632426649691190045388261930508, 9.698621357691645855154396779744
